# Questions tagged [billiards]

For questions about billiards; dynamical systems involving point particles (billiard balls) that travel in straight lines on the interior of some bounded domain (the billiard table) and experience perfectly elastic reflections on collision with boundary.

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### Arithmetic billiards, prime numbers and the Goldbach conjecture

On the online encyclopedia Wikipedia are edited the articles Arithmetic billiards and Dedekind psi function, the online encyclopedia Wolfram MathWorld has edited the article Goldbach Conjecture. We ...
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### Why a measure over the points of a billiard which reflect infinitely often in finite time vanish, i.e. why $\int_{N\cap M}f(x^{-})d\mu_1(x^{-1})=0$

The crux of my question is that why $\mu(N^{(2)}) = 0$ when $N^{(2)}$ consists all points of a billiard that reflect infinitely often in finite time under a given flow (i.e. transformation) function. ...
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### For arbitrary billiard tables with elastic boundary reflections, is the "Lebesgue measure" an invariant of the flow maps?

This question pertains to billiard dynamics and their invariant measures. Specifically, it concerns the oft-quoted 'fact' that billiard flow maps (built using specular reflection boundary conditions) ...
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### Particular values for the sum of divisors function from billiards

In this post we consider as reference the article Arithmetic billiards from Wikipedia. We consider the arithmetic billiard that is explained in the article. I've wondered if we can compute some simple ...
1 vote
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### Motivation for the definition of a compact Riemann manifold with piecewise smooth boundary

I am asking for either motivation on the requirement regarding $f_i^{-1}(0)$ in the following definition, or better yet a reference to a book dealing with this subject. The following is the definition ...
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### Plotting points which get further away when given the distance and amount of plot points

I am trying to mark up my pool table to run a cutting drill but I don't understand how to do the math which would give me the solution for my plot points. I have a total of 15 marks I need to make on ...
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### Using a simulation to show that any obtuse triangle whose largest angle is $\leq100^{\circ}$ has a stable periodic billiard orbit

In 2009, Richard Schwartz proved that any obtuse triangle whose largest angle is $\leq100^{\circ}$ has a stable periodic billiard orbit. My question then, is: How can I reproduce Schwartz's result ...
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### Proving a billiards bank shot procedure gives equal angles of incidence and reflection

One method of making a bank shot in billiards involves imagining two lines, the so called "cross-pocket" and "cross-ball" lines: One then projects their point of intersection to ...
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### Law of reflection when tangent is undefined

The law of reflection also holds for non-plane mirrors, provided that the normal at any point on the mirror is understood to be the outward pointing normal to the local tangent plane of the mirror at ...
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### Application of billiards

Studying billiards is a difficult problem in general, even in pretty simple cases it has plenty of interesting properties. I would like to understand what can be applications (mathematically or in ...
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### Rectangular billiards: ergodic or not?

I'm approaching dynamical billiards theory, and I've found that rectangular/square billiards are among the examples of integrable and non-ergodic ones. But in this contribution it is (resolutely) said ...
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### Sinai billiard isomorphic to hard spheres problem

I have heard that Sinai's famous results on the ergodicity of $n$ hard spheres defined on a $d$ dimensional hypercube with periodic boundaries is isomorphic to a billiard (from which it was proved), ...
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### Phase space and collision map of a billiard

I am working through Ch. 2 of Chaotic billiards by Chernov and Markarian. A few things are puzzling me about the basic construction and definitions. 2.5. Phase space for the flow. The state of a ...
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### Number of rays intersecting at a point inside a polygon

I'm working on a project to do with bouncing rays inside polygons and now I've reached a crucial stage of this project in which I need help with and is related to the problem stated below. Your help ...
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### Texts on Mathematical Billiards

I want to study the theory of mathematical billiards, and was looking for a text for self-study. If you have a text in mind that is more general i.e. on dynamical systems as a whole but still contains ...
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### Verifying that a billiard system satisfies the twist condition.

I'm numerically studying a 2D billiard system whose domain is a unit outer circle with an inner elliptical scatterer of variable geometry. EDIT: Both the circle and ellipse share a common centre. ...
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### Why are these angles equal?

-see picture above- (from Billiards and Geometry by Serge Tabachnikov) I don't understand why the angles $F_2BA_1$ and $F_1BA_0$ are equal (I do understand the conclusion, that follows from the ...
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### Billiards in a holey square

Suppose you start a point-billiard (or light ray) in a square at a random location, shooting off at a random angle, reflecting with angle-of-incidence equals angle-of-reflection. In general, because ...
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### Chaotic system?

I'm currently working on an individual project. In which I have created a program to simulate a particle bouncing around a finite bordered region. Where I need help in understanding how I might show ...
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### Joining boundary points through affine/billiards trajectories

Assume we have a smooth bounded domain $\Omega \subset \mathbb{R}^d$, and two points at the boundary $x, y$. I want to show that there exists an integer $n$ such that one can draw an affine path ...
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### Billiard ball mental patients.

The Question: Suppose there are $n$ extremely paranoid, vulnerable mental patients at a hospital. Each day at the lunch hour, they move around like frictionless billiard balls of radius $\rho$ metres ...
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### Unfolding a billiard trajectory on an unbounded billiard table

A common "trick" used in the study of polygonal billiards is to unfold the trajectory. I'm interested in billiards which may be bounded in one direction, and unbounded in another. For instance, ...
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### Reflecting a vector within a box (variation on the billiards problem)

Let there be a box, with bottom left corner at (0,0), and top right corner being at (m,n), where m and n are positive integers. A starting point is chosen at random within the box at (x,y), such ...
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### The simplest billiards problem

The Problem: Let's say we have a rectangle of size $m \times n$ centered at the origin (or, if it makes the math easier, you can place it wherever on the plane). We take a billiard ball, ...
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### Simple Billiard Table with Positive Lyapunov exponent

I can understand the Lyapunov exponent for a circular billiard table is zero. However, if I were right, I read the Lyapunov exponent for a stadium billiard table is non-zero which was surprising for ...
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### Trajectories in circular Billards

Given two points $p_1,p_2$ on a circular billard table. I want to know all billard trajectories from $p_1$ to $p_2$ hitting the boundary precisely once. Model of the circular billard: Denote the ...
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